1. Quantum mechanics II Winter 2011
Physics 452
Quantum mechanics II
Winter 2011
Karine Chesnel

2. Homework Phys 452 Today Apr 1: assignment #21 11.5, 11.6, 11.7
Wed Apr 6: assignment #22
11.8, 11.10, 11.11, 11.13
Friday April 8: assignment #23
11.14, 11.18, Compton question
Sign up for the QM & Research presentations
Next week, W April 6 or F April 8
Homework #24
20 pts

3. Treats and vote for best presentation
Phys 452
Class- schedule
Today April 1st: Born approximation
Monday 4th : Born approx., Compton effect
Wednesday 6th : research & QM presentations I
Friday 8th :research & QM presentations II
Treats and vote for best presentation
In each session

4. Research and QM presentation
Phys 452
Research and QM presentation
Template
My cool
research!
…or doing simulations
or theory
As an experimentalist
In the lab …

5. Research and QM presentation
Phys 452
Research and QM presentation
Template
Focus on
one physical
principle or phenomenon
involved
in your research
Make a connection
with a topic covered in
Quantum Mechanics:
A principle
An equation
An application

6. Scattering Phys 452 Quantum treatment Spherical wave Plane wave
Relationship
with cross-section

7. in the scattering region
Phys 452
Scattering
Partial wave analysis
Connecting all three regions and expressing the
Global wave function in spherical coordinates
Rayleigh’s formula
Scattered waves
To be determined
by solving the
Schrödinger equation
in the scattering region
+ boundary conditions
Total
cross-section

8. Scattering Phys 452 Phase - shifts Scattering amplitude Scattering
Cross-section

9. Scattering – phase shift
Phys 452
Scattering – phase shift
Pb 11.5
Reflection against a wall
1) Solve the Schrödinger equation
In region 1
In region 2
2) Continuity at boundary:
3) Identify the phase shift
wall
Region 1
Region 2

10. Scattering – Phase shift
Phys 452
Scattering – Phase shift
Pb Hard sphere scattering
(Pb 11.3)
Boundary conditions
We found
Express the phase shift: using
Express in terms of functions and

11. Scattering- phase shifts
Phys 452
Scattering- phase shifts
Pb 11.7
Spherical delta function shell (Pb 11.4)
Do NOT do the assumption
Outside:
Inside:
Boundary conditions
Continuity of
Discontinuity of
Express in terms of and

12. Born formalism Phys 452 Worked together with Albert Einstein
(Nobel Prize 1921
Photoelectric effect)
Werner Heisenberg
(Nobel Prize 1932
Creation of QM)
Max Born ( )
German physicist
Nobel prize in 1954
For interpretation of probability of density y

13. What is the main idea of the Born approximation?
Phys 452
Quiz 32
What is the main idea of the Born approximation?
To develop a formalism where we express the wave function
in terms of Green’s functions
B. To use Helmholtz equation instead of Schrödinger equation
C. To find an approximate expression for y when far away from
the scattering center
D. To express the scattering factor in terms of scattering vector
E. To find the scattering factor in case of low energy

14. Scattering Phys 452 Quantum treatment q Spherical wave Plane wave
Easy formula to calculate f(q,f)?
or f(q)?

15. Born formalism Phys 452 Green’s function Schrödinger equation
Helmholtz equation
Helmholtz
Solution
Green’s
function
George Green
British Mathematician

16. Phys 452
Born formalism
Using Fourier Transform of Helmholtz equation
and contour integral with Cauchy’s formula, one gets:
Green’s function
Pb 11.8
Check that it satisfies the Helmholtz equation
Integral form of the Schrödinger equation

17. Born approximation Phys 452 Spatial approximation
The incoming wave is scattered in a wave of same amplitude,
just different direction
(First order) Born approximation

18. Born approximation Phys 452 (First order) Born approximation
Scattering vector

19. Born approximation Phys 452 Examples: Soft-sphere Yukawa potential
Low energy approximation
Examples:
Soft-sphere
Case of spherical potential
Yukawa potential
Rutherford
scattering

20. Born approximation Phys 452 Pb 11.10 Soft sphere potential
Scattering amplitude
Case of spherical potential
Approximation at low E
Develop and to third order

21. Scattering – Phase shift
Phys 452
Scattering – Phase shift
Pb Yukawa potential
Expand

22. Scattering- phase shifts
Phys 452
Scattering- phase shifts
Pb 11.13
Spherical delta function shell (Pb 11.4)
Low energy case
More general spherical potential